Problem Statement
Let $a_n$ be an infinite sequence of positive integers such that $\sum \frac{1}{a_n}$ converges. There exists some integer $t\geq 1$ such that\[\sum \frac{1}{a_n+t}\]is irrational.
Categories:
Irrationality
Progress
This conjecture is due to Stolarsky.A negative answer was proved by Kovač and Tao [KoTa24], who proved even more: there exists a strictly increasing sequence of positive integers $a_n$ such that\[\sum \frac{1}{a_n+t}\]converges to a rational number for every $t\in \mathbb{Q}$ (with $t\neq -a_n$ for all $n$).
Source: erdosproblems.com/266 | Last verified: January 14, 2026